Method of creating particle size distribution model, method of predicting degradation of fuel cell catalyst using the method of creating particle size distribution model, and method of controlling fuel cell using the method of predicting degradation of fuel cell catalyst

ABSTRACT

A particle size distribution creating method includes a particle size range determining step, an integrating step of integrating the frequency of appearance of particles within the particle size range determined in the particle size range determining step, a division point determining step of determining particle sizes that provide division points, using the integral of the frequency of appearance obtained in the integrating step, and a typical point determining step of determining the minimum particle size, maximum particle size and the particle sizes of the division points as typical points. This method is characterized by assuming a particle size distribution which contains particles having the particle sizes of the respective typical points and is plotted such that the frequency of appearance of the particles having the particle size of each of the typical points is equal to the integral over each of the regions defined by the typical points, and obtaining the assumed particle size distribution as a particle size distribution model.

INCORPORATION BY REFERENCE

The disclosure of Japanese Patent Application No. 2009-084428 filed onMar. 31, 2009 including the specification, drawings and abstract isincorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The invention relates to a method of creating a particle sizedistribution model, a method of predicting degradation of a fuel cellcatalyst, using the method of creating the particle size distributionmodel, and a method of controlling a fuel cell, using the method ofpredicting degradation of the fuel cell catalyst.

2. Description of the Related Art

Fuel cells are operable to convert chemical energy directly intoelectrical energy, by supplying fuel and an oxidizing agent to twoelectrodes that are electrically connected, and electrochemicallycausing oxidation of the fuel. Unlike thermal power generation, fuelcells are free from constraints of the Carnot cycle, and thus exhibit ahigh energy conversion efficiency. A fuel cell generally consists of aplurality of single cells laminated or stacked together, and each of thesingle cells has a basic structure in the form of a membrane electrodeassembly in which an electrolyte membrane is sandwiched by and between apair of electrodes. In particular, a polymer electrolyte fuel cell usinga solid polymer electrolyte membrane as the electrolyte membrane has theadvantages of being easily reduced in size and operating at lowtemperatures, and is therefore noteworthy for its use as a portablepower supply or a power supply for a mobile unit.

In the polymer electrolyte fuel cell, a reaction of the followingformula (I) proceeds at the anode (fuel electrode) when hydrogen is usedas the fuel.

H ₂→2H ⁺+2e ⁻  (I)

Electrons generated in the reaction of the above formula (I) passthrough an external circuit, do work at an external load, and then reachthe cathode (oxidant electrode). Protons generated in the reaction ofthe above formula (I) transfer by electrical permeation from the anodeside to the cathode side in the solid polymer electrolyte membrane whilethey are in a hydrated state.

Also, a reaction of the following formula (II) proceeds at the cathodewhen oxygen is used as the oxidizing agent.

2H ⁺+(½)O ₂₊₂ e ⁻ →H ₂ O  (II)

Water formed at the cathode passes mainly though a gas diffusion layer,and is discharged to the outside. Thus, the fuel cell is a clean powergenerator since nothing but water is discharged from the fuel cell.

FIG. 9 schematically shows a cross-section of a single cell 100 of ageneral polymer electrolyte fuel cell when it is cut in a direction oflamination of layers. The single cell 100 includes a membrane electrodeassembly 8 consisting of a solid polymer electrolyte membrane (which maybe simply called an electrolyte membrane) 1 having hydrogen ionconductivity, and a cathode 6 and an anode 7 between which theelectrolyte membrane 1 is sandwiched. The single cell 100 furtherincludes separators 9 and 10 located outwardly of the electrodes (i.e.,the cathode 6 and anode 7), respectively. The membrane electrodeassembly 8 is sandwiched by and between the separators 9 and 10. Gaschannels 11 and 12 are formed at the boundaries of the separators andthe electrodes, and hydrogen gas is continuously supplied to the anode,while gas (normally, air) containing oxygen is continuously supplied tothe cathode. Generally, each electrode consists of a catalyst layer anda gas diffusion layer, which are laminated in this order as viewed fromthe electrolyte membrane. Namely, the cathode 6 consists of a cathodecatalyst layer 2 and a gas diffusion layer 4 that are laminated on eachother, and the anode 7 consists of an anode catalyst layer 3 and a gasdiffusion layer 5 that are laminated on each other.

One of the problems encountered in the polymer electrolyte fuel cell isvoltage reduction caused by dissolution of catalyst metal in theelectrodes. With regard to this problem, a mathematical model thatsimulates oxidation and dissolution of a platinum catalyst when it isused as a catalyst metal is discussed, and calculation results usingthis model are described in a non-patent document (R. M. Darling and J.P. Meyers: J. Electrochem. Soc., vol. 150, pages A1523-A1527, 2003).

In the above-identified non-patent document, the rates of reactions,i.e., oxidation and dissolution, of the platinum catalyst, arespecifically discussed. However, even if the mathematical modeldescribed in this document is used, precise simulation results thatagree with experimental results are not necessarily obtained, as isapparent from FIG. 1 and FIG. 5 of this document.

SUMMARY OF THE INVENTION

The invention provides a method of creating a particle size distributionmodel with improved preciseness, within a shortened calculation time, amethod of predicting degradation of a fuel cell catalyst, using themethod of creating the particle size distribution model, and a method ofcontrolling a fuel cell, using the method of predicting degradation ofthe fuel cell catalyst.

A first aspect of the invention is concerned with a method of creating aparticle size distribution model that simulates a particle sizedistribution of a cluster of particles of a catalyst metal of a fuelcell, which includes a plurality of particles of the catalyst metal. Theparticle size distribution model creating method includes the steps of:determining a particle size range by determining a minimum particle sizeand a maximum particle size of the cluster of particles of the catalystmetal to be simulated, integrating the frequency of appearance of theparticles in the determined particle size range, over an integrationregion defined by the minimum particle size as a starting point and themaximum particle size as an endpoint, dividing the integration regioninto a given number of regions through a first dividing operation, usingthe integral of the frequency of appearance, so that integrals obtainedfor the individual regions into which the integration region is dividedare substantially equal, and determining particles sizes of divisionpoints at which the integration region is divided, determining theminimum particle size, the maximum particle size and the particle sizesof the respective division points, as typical points, and obtaining aparticle size distribution model by assuming a particle sizedistribution containing particles having the particle sizes of therespective typical points, the particle size distribution being plottedsuch that the frequency of appearance of the particles having theparticle size of each of the typical points is equal to the integralobtained for each of the regions into which the integration region isdivided at the typical points.

In the particle size distribution model creating method as describedabove, the particle size distribution is determined, using the minimumparticle size, the maximum particle size, and the respective points atwhich the integral of the frequency of appearance is equally or evenlydivided, as typical points. It is thus possible to create a particlesize distribution model that can precisely simulate a particle sizedistribution obtained by experiment, with a reduced number of variables,as compared with the particle size distribution model of the relatedart.

A second aspect of the invention is concerned with a method ofpredicting degradation of a catalyst metal of a fuel cell. Thedegradation predicting method includes the steps of: determining aparticle size range by determining a minimum particle size and a maximumparticle size of a cluster of particles of the catalyst metal of which aparticle size distribution is to be simulated, the cluster of particlesincluding a plurality of particles of the catalyst metal, integratingthe frequency of appearance of the particles in the determined particlesize range, over an integration region defined by the minimum particlesize as a starting point and the maximum particle size as an endpoint,dividing the integration region into a given number of regions through afirst dividing operation, using the integral of the frequency ofappearance, so that integrals obtained for the individual regions intowhich the integration region is divided are substantially equal, anddetermining particles sizes of division points at which the integrationregion is divided, determining the minimum particle size, the maximumparticle size and the particle sizes of the respective division points,as typical points, obtaining a particle size distribution model byassuming a particle size distribution containing particles having theparticle sizes of the respective typical points, the particle sizedistribution being plotted such that the frequency of appearance of theparticles having the particle size of each of the typical points isequal to the integral obtained for each of the regions into which theintegration region is divided at the typical points, and predictingdegradation of the catalyst metal of the fuel cell, using the particlesize distribution model.

According to the method of predicting degradation of the fuel cellcatalyst as described above in which the above-described particle sizedistribution model creating method is used for prediction of degradationof the fuel cell catalyst, degradation of the fuel cell catalyst whichis more likely to occur in reality can be precisely simulated within thesame calculation lime, as compared with the case where the particle sizedistribution model of the related art is used for prediction ofdegradation of the fuel cell catalyst. Also, according to the method ofpredicting degradation of the fuel cell catalyst of the invention inwhich the above-described particle size distribution model creatingmethod is used for prediction of degradation of the fuel cell catalyst,it is possible to provide results having the same or equivalentpreciseness as results obtained when the particle size distributionmodel of the related art is used for prediction of degradation of thefuel cell catalyst, within a shorter time than that required in therelated art.

A third aspect of the invention is concerned with a method ofcontrolling a fuel cell. The control method includes the steps of:determining a particle size range by determining a minimum particle sizeand a maximum particle size of a cluster of particles of the catalystmetal of which a particle size distribution is to be simulated, thecluster of particles including a plurality of particles of the catalystmetal, integrating the frequency of appearance of the particles in thedetermined particle size range, over an integration region defined bythe minimum particle size as a starting point and the maximum particlesize as an endpoint, dividing the integration region into a given numberof regions through a first dividing operation, using the integral of thefrequency of appearance, so that integrals obtained for the individualregions into which the integration region is divided are substantiallyequal, and determining particles sizes of division points at which theintegration region is divided, determining the minimum particle size,the maximum particle size and the particle sizes of the respectivedivision points, as typical points, obtaining a particle sizedistribution model by assuming a particle size distribution containingparticles having the particle sizes of the respective typical points,the particle size distribution being plotted such that the frequency ofappearance of the particles having the particle size of each of thetypical points is equal to the integral obtained for each of the regionsinto which the integration region is divided at the typical points,predicting degradation of the catalyst metal of the fuel cell, using theparticle size distribution model, measuring a cell voltage of the fuelcell, measuring a cell resistance of the fuel cell, determining whetherhumidity control of the fuel cell is to be performed, based on a degreeof the predicted degradation of the catalyst metal of the fuel cell, themeasured cell voltage, and the measured cell resistance, and selectingand executing a first control mode in which control for reducing thehumidity of the fuel cell is performed if the measured cell resistanceis smaller than a predetermined resistance value, a second control modein which control for increasing the humidity of the fuel cell isperformed if the measured cell resistance is equal to or larger than thepredetermined resistance value, and a third control mode in whichcontrol for increasing the humidity of the fuel cell is performed if itis predicted, from a result of degradation prediction of the catalystmetal of the fuel cell, that a surface area of a local portion of thefuel cell catalyst is reduced.

According to the present invention, the particle size distribution isdetermined, using the minimum particle size, the maximum particle size,and the respective points at which the integral of the frequency ofappearance is equally or evenly divided, as typical points. It is thuspossible to create a particle size distribution model that can preciselysimulate a particle size distribution obtained by experiment, with areduced number of variables, as compared with the particle sizedistribution model of the related art.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and further features and advantages of the invention willbecome apparent from the following description of example embodimentswith reference to the accompanying drawings, wherein like numerals areused to represent like elements and wherein:

FIG. 1A is a schematic view specifically showing a division pointdetermining step of the invention, and FIG. 1B is a graph schematicallyshowing a particle size distribution model created by a creating methodof the invention;

FIG. 2 is a graph schematically showing a cumulative distributionfunction F(X);

FIG. 3 is a graph schematically showing a simulation result in a firstapplication of a method of predicting degradation of a fuel cellcatalyst of the invention, wherein the horizontal axis indicates theoperating time, and the vertical axis indicates the platinum catalystsurface area;

FIG. 4 is a graph showing examples of calculation results indicatingprediction results obtained in a fuel cell catalyst degradationpredicting step of the invention;

FIG. 5 is a flowchart illustrating a typical example of a method ofcontrolling a fuel cell according to the invention;

FIG. 6A is a schematic view of a platinum catalyst particle sizedistribution model used in an embodiment of the invention, and FIG. 6Bis a graph indicating the platinum surface area maintenance factor withrespect to the operating time (the number of power generation cycles);

FIG. 7 is a graph schematically showing a platinum catalyst particlesize distribution, which is simplified and taken into a calculationmodel in the related art;

FIG. 8A is a schematic view of a platinum catalyst particle sizedistribution model used in a comparative example, and FIG. 8B is a graphindicating the platinum surface area maintenance factor with respect tothe operating time (the number of power generation cycles); and

FIG. 9 is a view schematically showing a cross-section of a single cellof a general polymer electrolyte fuel cell when cut in a direction oflamination of layers.

DETAILED DESCRIPTION OF EMBODIMENTS

A particle size distribution model creating method of the invention,which is a method of creating a particle size distribution model thatsimulates a particle size distribution of a cluster of particles as acollection of a plurality of particles having different particle sizes,includes a particle size range determining step of determining theminimum particle size and maximum particle size of the particlesincluded in the cluster of particles to be simulated, an integratingstep of integrating the frequency of appearance of the particles withinthe particle size range determined in the particle size rangedetermining step, over an integration region defined by the minimumparticle size as a starting point and the maximum particle size as anendpoint, a division point determining step of dividing the integrationregion used in the integrating step into a given number of regionsthrough a first dividing operation, using the integral of the frequencyof appearance obtained in the integrating step, so that the integralsover the individual regions into which the integration region is dividedare substantially equal, and determining particles sizes of divisionpoints at which the integration region is divided, and a typical pointdetermining step of determining the minimum particle size, the maximumparticle size and the particle sizes of the respective division points,as typical points. The model creating method is characterized byassuming a particle distribution which contains particles having theparticle sizes of the respective typical points, and is plotted suchthat the frequency of appearance of the particles having the particlesize of each of the typical points is equal to the integral obtained foreach of the regions into which the integration region is divided at thetypical points, and obtaining the assumed particle size distribution asa particle size distribution model.

Some examples of techniques for measuring a particle size distributionof a platinum catalyst by experiment have been reported in the relatedart. For example, an article of H. A. Gasteiger et al.: J. Electrochem.Soc., vol. 152, pages A2256-A2271, 2005 (which will be called “Article1”) discusses measuring the surface area of particles of the platinumcatalyst as well as measuring the particle size distribution of theplatinum catalyst. In Article 1, a graph (FIG. 7 of Article 1) showingthe actually measured particle size distribution of the platinumcatalyst in a cathode of a membrane electrode assembly, and a graph(FIG. 4 of Article 1) showing measurement results of the surface area ofthe platinum catalyst are provided. The graph of FIG. 7 of Article 1, inwhich the horizontal axis indicates the catalyst particle size and thevertical axis indicates the number of particles, shows measurementresults of particle sizes of 200 spherical particles selected fromplatinum catalyst particles in the initial state of the fuel cell, and200 spherical particles selected from platinum catalyst particles afterpower generation. All of the particles sizes of the selected particleswere measured through observation with a transmission electronmicroscope (TEM). The particle size distribution of the platinumcatalyst particles (the average particle size=5.9 nm) after powergeneration has a smaller height and a larger width, in other words, isin a broad condition, as compared with the particle size distribution ofthe platinum catalyst particles (the average particle size=2.8 nm) ofthe initial state. This phenomenon occurs because dissolution of theplatinum catalyst proceeds as the operating time of the fuel cellpasses, resulting in a further reduction of the particles sizes ofrelatively small particles due to the dissolution, and the agglomerationof dissolved platinum and particles having relatively large particlesizes occurs, resulting in a further increase in the particle size. Inthe graph of FIG. 4 of Article 1, the horizontal axis indicates thenumber of power generation cycles, and the vertical axis indicates atotal surface area of the catalyst particles. As is understood from FIG.7 as described above, the dissolution of the platinum catalyst causesthe agglomeration of dissolved platinum and particles having relativelarge particle sizes; therefore, the total surface area of the catalystparticles decreases as the number of power generation cycles increases.As the total surface area of the catalyst particles decreases, acatalyst reaction field that governs a reaction at the electrode of thefuel cell is reduced, resulting in degradation of the fuel cellcatalyst.

If the particle size distribution of the platinum catalyst obtained bymeasurement/experiment as described above is used for predictingdegradation of the fuel cell catalyst, simulation results cannot beobtained within a practical calculation time. This is because theparticle size distribution of platinum particles is actually acontinuous distribution, and therefore, an enormously large number ofdata points are required to achieve precise simulation.

In the meantime, an article of W. Bi and T. F. Fuller: J. Power Sources,vol. 18, pages 188-196, 2008 (which will be called “Article 2”)discusses simplifying a platinum catalyst particle size distribution,and incorporating it into a calculation model. FIG. 7 is a graphschematically, showing the platinum catalyst particle size distributiondiscussed in Article 2. In the graph of FIG. 7, the horizontal axisindicates the particle size, and the vertical axis indicates the numberof platinum atoms. As is understood from FIG. 7, the platinum catalystparticle distribution model discussed in Article 2 is represented by twotypes of platinum particles having different particle sizes, withouttaking account of changes in the number of particles. In Article 2, agraph (FIG. 3 of Article 2) is provided in which the horizontal axisindicates the operating time and the vertical axis indicates the totalsurface area of catalyst particles. As is understood from FIG. 3 ofArticle 2, two graphs (graphs indicated by solid lines in FIG. 3 ofArticle 2) indicating simulation results using the model as shown inFIG. 7 do not match or agree with a graph (graph indicated by a brokenline in FIG. 3 of Article 2) indicating results obtained by experiment.This is because the surface area of the platinum particles does notchange continuously, due to the use of the particle size distributionmodel represented by two types of platinum particles having differentparticle sizes without taking account of changes in the number ofparticles.

FIG. 8A is a graph schematically showing a particle size distributionmodel as an application of the platinum catalyst particle sizedistribution discussed in Article 2. In FIG. 8A, the vertical axis andhorizontal axis indicate the same parameters as those of FIG. 7. As isunderstood from FIG. 8A, the particle size distribution model of FIG. 8Asimulates the actual particle size distribution (indicated by a curve inFIG. 8A) as shown in FIG. 7 of Article 1. In the particle sizedistribution model (vertical straight lines in FIG. 8A), a number ofparticle sizes in the particle size distribution are selected, such thatthe particle sizes are spaced at equal intervals on the horizontal axis,and the number of particles having each particle size changes along thehorizontal axis. FIG. 8B is a graph in which the horizontal axisindicates the operating time, and the vertical axis indicates the totalsurface area of catalyst particles. As is understood from FIG. 8B, agraph (solid line) indicating simulation results using the model shownin FIG. 8A does not match or agree with plots indicating resultsobtained by experiment. Accordingly, it is found difficult to simulatethe results obtained by experiment, with the particle size distributionmodel as a mere application of the related art. Thus, the known methodscannot reproduce an exponential reduction of the platinum catalystsurface area, which is observed, by experiment over a long period ofoperating time of the fuel cell. The present invention provides a methodof creating a precise particle size distribution model within a reducedcalculation time, using differences of particles sizes of differentparticles as variables, unlike the particle size distribution modelcreating method of the related art as described above.

The “frequency of appearance” mentioned above in relation to theinvention may be represented by the number of particles, mass, orvolume. When the frequency of appearance is represented by the number ofparticles, the particle size distribution becomes a left-rightsymmetrical distribution (for example, a normal distribution), as shownin FIG. 7 of the above-described Article 1. When the frequency ofappearance is represented by mass or volume, the particle sizedistribution is not a left-right symmetrical, normal distribution, butan asymmetrical distribution, since the mass or volume generallyincreases as the particle size increases. However, such a particle sizedistribution does not cause any problem when a long-term operation issimulated, even though slight errors or deviations from experimentalvalues appear in the initial period of the operating time.

The method of creating a particle size distribution model according tothe invention includes at least the particle size range determiningstep, integrating step, division point determining step, and the typicalpoint determining step. The particle size range determining step to beperformed first is a step of determining the minimum particle size andmaximum particle size of particles included in a cluster of particles tobe simulated. In this step, the average particle size as well as theminimum particle size and the maximum particle size may be determined.

The integrating step to be performed subsequent to the particle sizerange determining step is a step of integrating the frequency ofappearance of particles within the particle size range determined in theparticle size range determining step, over an integration region fromthe minimum particle size as a starting point to the maximum particlesize as an end point. In the integrating step, the sum of thefrequencies of appearance of all of the particles in the particle sizerange determined in the particle size range determining step can becalculated.

The division point determining step to be performed subsequent to theintegrating step is a step of dividing the integration region used inthe integrating step into a given number of regions through a firstdividing operation, using the integral of the frequency of appearanceobtained in the integrating step, so that integrals obtained for theindividual regions into which the integration region is divided areequal. FIG. 1A is a schematic view specifically showing the divisionpoint determining step of the invention. More specifically, FIG. 1A is agraph in which the vertical axis indicates the frequency f(x) ofappearance, and the horizontal axis indicates the particle size x, whilea curve 20 represents the actual particle distribution (normaldistribution). When the division point determining step is executed, theminimum particle size x_(min), and maximum particle size x_(max) ofparticles included in the cluster of particles to be simulated havealready been determined in the above-described particle size rangedetermining step. Also, the integral (the area of a range defined by thecurve 20 and the horizontal axis) has already been determined in theabove-described integrating step, by integrating the frequency ofappearance of the particles over the range from the minimum particlesize x_(min) as a starting point to the maximum particle size x_(max),as an end point. In FIG. 1A, the integration region is divided into 10regions through the first dividing operation, using the above-mentionedintegral, so that area a=area b=area c= . . . =area i=area j in FIG. 1A.In this manner, division points X₁, x₂, . . . x₉ are determined so thatthe integrals obtained for the individual regions defined by thedivision points are equal. In the typical point determining step as thelast step, the particle sizes of the division points x₁, x₂, . . . x₉,and the minimum particle size x_(min) and the maximum particle size aredetermined as typical points.

In the division point determining step, a method using a probabilitydensity function as indicated below, for example, may be used as amethod of actually determining the division points x₁, x₂, . . . x₉. Inthe following, an example in which the method is applied to a normaldistribution will be described. The normal distribution has aprobability density function as expressed by the following equation (1).In the equation (1), x denotes a probability variable, μ denotes anaverage value, and σ denotes a variance.

$\begin{matrix}{{{F(x)} = {\frac{1}{\sqrt{2{\pi\sigma}^{2}}}{\exp \left( \frac{- \left( {x - \mu} \right)^{2}}{2\sigma^{2}} \right)}}},{{- \infty} < x < \infty},{\sigma^{2} > 0}} & (1)\end{matrix}$

A cumulative distribution function F(X) using the probability densityfunction f(x) is represented by the following equation (2).

$\begin{matrix}{{F(X)} = {\int_{- \infty}^{x}{{f(x)}\ {x}}}} & (2)\end{matrix}$

When an inverse function of the cumulative distribution function F(X) isdenoted as F⁻¹ (X)=G(Y) (=X), values of X₁, X₂, . . . X_(n) are obtainedwith respect to values of Y₁, Y₂, . . . Y_(n), by using the inversefunction. The values of Y₁, Y₂, . . . Y_(n) may be determined, forexample, in the following manner. FIG. 2 is a graph schematicallyshowing the cumulative distribution function F(X). By placing Y₁, Y₂, .. . Y_(n) at equal intervals on the F(X) axis, X₁, X₂, . . . X_(n)corresponding to Y₁. Y₂, . . . Y_(n) can be obtained. The X_(i), X₂, . .. X_(n) obtained through this operation can be adopted as divisionpoints.

In fact, it is extremely difficult to obtain the inverse function bymanual calculation. On the other hand, the inverse function of thecumulative distribution function F(X) may be easily obtained bysoftware, such as spreadsheet software. For example, with the use ofMicrosoft Office Excel (trade name, manufactured by Microsoft) as onetype of spreadsheet software, values of X_(r), X₂, . . . X_(n) can beobtained by using a NORMSINV function that returns values of the inversefunction of the cumulative distribution function of a, standard normaldistribution, or a NORMINV function that returns values of the inversefunction of the cumulative distribution function of a normaldistribution with respect to the designated average and standarddeviation.

Considering that the more precise particle size distribution can bereproduced by dividing a part of the region or the entire region intofurther smaller regions, one form of the particle size distributionmodel creating method of the invention may be configured such that, inthe division point determining step, a part of or all of the givennumber of regions into which the determined particle size range isdivided through the first dividing operation is divided further into agiven number of regions, such that the integrals obtained for theindividual regions resulting from the second dividing operation areequal, and the particle sizes that provide the division points in thefirst and second dividing operations are determined. In particular, asone form of the particle size distribution model creating method of theinvention, which is useful for investigating changes in the particlesize distribution over a long period of time, the second dividingoperation may be performed on a region including the maximum particlesize, out of the given number of regions into which the determinedparticle size range is divided through the first dividing operation, forthe reasons as follows. As described above with reference to FIG. 7 ofArticle 1, particles having relatively small particle sizes dissolve ina very short time, and the agglomeration of these particles and otherparticles occurs; therefore, it is not necessary to keep track of thebehaviors of these particles particularly in detail when changes in theparticle size distribution over a considerably long period of time arepredicted. On the other hand, it is particularly important to keep trackof the behaviors of particles having relatively large particle sizeswhen changes in the particle size distribution over a considerably longperiod of time are predicted. However, as shown in FIG. 1B as describedabove, only one typical point exists, in particular, in the regionincluding the maximum particle size, which is not necessarily sufficientto keep track of detailed changes in the particle size distribution.Accordingly, the region including the maximum particle size (the regionhaving area j in FIG. 1A) is further divided into a given number ofregions so that integrals for the individual regions are equal, therebyto provide the particle size distribution model useful for investigatingchanges in the particle size distribution over a long period of time.

FIG. 1B is a graph schematically showing a particle size distributionModel created by the creating method of the invention. In the graph ofFIG. 1B, the vertical axis indicates the frequency of appearance, andthe horizontal axis indicates the particle size x. The particle sizedistribution model shown in FIG. 1B contains particles having each ofthe particle sizes x_(i), x₂, . . . x₉ determined in the typical pointdetermining step, and the minimum particle size x_(min) and the maximumparticle size x_(max), and assumes a particle size distribution in whichthe frequency of appearance of particles having the particle size ofeach typical point is equal to the integral S of each of the regionsdefined by the typical points. Thus, the points at which the integral ofthe frequency of appearance is equally divided, the minimum particlesize and the maximum particle size are used as the typical points;therefore, the particle size distribution model created by the particlesize distribution model creating method according to the invention canprecisely simulate the particle size distribution obtained byexperiment, with a reduced number of variables, as compared with theparticle size distribution model of the related art.

As one form of the particle size distribution model creating method ofthe invention, the particle size distribution may be in the form of anormal distribution since the particle size distribution can be moreprecisely reproduced by use of the normal distribution. It is, however,to be understood that the particle size distribution is not limited tothe normal distribution, but various probability distributions may beused provided that the actual particle size distribution can beprecisely reproduced. While the types of probability distributionsinclude discrete distribution type, absolute discrete distribution type,and continuous distribution type, it is preferable to use a probabilitydistribution of continuous type since one of the main objects of theinvention is to predict degradation of a fuel cell catalyst, and theactual distribution of catalyst particles is a continuous distribution.Examples of the probability distribution of continuous type include, forexample, a logarithmic normal distribution, exponential distribution,t-distribution, chi-square distribution, gamma distribution, betadistribution, F-distribution, Cauchy distribution, Erlang distribution,triangular distribution, Laplace distribution, Rayleigh distribution,logistic distribution, Pareto distribution, Weibull distribution, andfunctions referring to the actually measured platinum particle sizedistribution.

A method of predicting degradation of the fuel cell catalyst accordingto the invention is characterized by including a step of creating aparticle size distribution model of the fuel cell catalyst, using theabove-described particle size distribution model, and a step ofpredicting degradation of the fuel cell catalyst, using the particlesize distribution model of the fuel cell catalyst.

The method of predicting degradation of the fuel cell catalyst of theinvention has the particle size distribution model creating step and thestep of predicting degradation of the fuel cell catalyst (which will becalled “fuel cell catalyst degradation predicting step”). Of thesesteps, the particle size distribution model creating step has alreadybeen described in the explanation of the particle size distributionmodel creating method of the invention.

In the fuel cell catalyst degradation predicting step according to theinvention, the above-described particle size model creating method isused for prediction of degradation of the fuel cell catalyst; therefore,degradation of the fuel cell catalyst which is more likely to occur inreal operating situations can be precisely simulated within the samecalculation time, as compared with the case where the particle sizedistribution model of the related art is used for prediction ofdegradation of the fuel cell catalyst Also, in the fuel cell catalystdegradation predicting step of the invention, the use of theabove-described particle size distribution model creating method forprediction of degradation of the fuel cell catalyst makes it possible toyield results having substantially the same degree of preciseness asresults provided when the particle size distribution model of therelated art is used for prediction of degradation of the fuel cellcatalyst, within a shorter time than that of the related art.

In one form of the fuel cell catalyst degradation predicting method ofthe invention, at least one mathematical model selected from amathematical model indicative of the rate of platinum dissolutionreaction, a mathematical model indicative of the rate of platinumoxidation reaction, and a mathematical model indicative of materialbalance may be used. Of the above-indicated mathematical models, all ofthe mathematical model indicative of the rate of platinum dissolutionreaction, mathematical model indicative of the rate of platinumoxidation reaction, mathematical model indicative of the rate ofdissolution of platinum oxide (II) and the mathematical model indicativeof material balance may be used in the fuel cell catalyst degradationpredicting step.

Table 1 below shows the meanings of symbols used in the mathematicalmodels used in the fuel cell catalyst degradation predicting step of theinvention.

TABLE 1 Symbol Unit Meaning C_(H+) mol/l proton concentrationC_(H+, ref) — coefficient conversion constant of C_(H+) 1 mol/l→mol/cm³(1/1000) C_(Pt2+) mol/l Pt ion concentration C_(Pt2+, ref) — coefficientconversion constant of C_(Pt2+) 1 mol/l→mol/cm³( 1/1000) D_(Pt2+)cm²/sec diffusion coefficient of Pt ions (1E−6 cm²/sec, from Document 2)F C/equiv Faraday constant (96485 C equiv⁻¹) k₁ mol/(cm²sec) rateconstant of Pt dissolution k₂ mol/(cm²sec) rate constant of Pt oxidationk₃ mol/(cm²sec) rate constant of PtO dissolution M_(Pt) g/mol weight ofPt atom (195 g/mol) M_(PtO) g/mol weight of PtO atom (211.09 g/mol) n₁equiv/mol number of electrons involved in Pt dissolution (2 equiv/mol)n₂ equiv/mol number of electrons involved in Pt oxidation (2 equiv/mol)R J/(mol · K) gas constant (8.314 J/(mol · K)) R(i, z) cm Pt particleradius T K temperature U₁ V thermokinetic reversible potential of Ptdissolution U₂ V thermokinetic reversible potential of Pt oxidationU^(θ) ₁ V standard thermokinetic potential of Pt dissolution U^(θ) ₂ Vstandard thermokinetic potential of Pt oxidation α_(a, 1) — anodictransfer coefficient of Pt dissolution α_(a, 2) — anodic transfercoefficient of Pt oxidation α_(c, 1) — cathodic transfer coefficient ofPt dissolution α_(c, 2) — cathodic transfer coefficient of Pt oxidationρ_(Pt) g/cm³ density of Pt (21.95 g/cm³) ρ_(PtO) g/cm³ density of PtO(14.1 g/cm³) ω J/mol PtO—PtO interaction coefficient θ_(νBC) —proportion of Pt surface not covered with oxide θ_(PtO) — proportion ofPt surface covered with oxide σ_(Pt) J/cm² surface tension of Ptparticles σ_(PtO) J/cm² surface tension of PtO particles E V cellpotential Δμ⁰ _(PtO) J/mol chemical potential shift of PtO

The following equation (3) may be used as a mathematical modelindicative of the rate of platinum dissolution reaction.

$\begin{matrix}{{r_{1}\left( {,z} \right)} = {k_{1}{{\theta_{vac}\left( {i,z} \right)}\begin{bmatrix}{{\exp \left( {\frac{\alpha_{n,1}n_{1}F}{RT}\left( {E - {U_{1}\left( {i,z} \right)}} \right)} \right)} -} \\\left( \frac{C_{{Pt}\;,{2 +}}(z)}{C_{{Pt},{2 +},{ref}}} \right) \\{\exp\left( {\frac{\alpha_{c,1}n_{1}F}{RT}\left( {E - {U_{1}\left( {U_{1}\left( {i,z} \right)} \right)}} \right)} \right.}\end{bmatrix}}}} & (3)\end{matrix}$

In the above equation (3), i is the number of types of particles havingdifferent radii (the same definition applies to the equations below).The potential U₁(i, z) in the above equation (3) is defined by thefollowing equation (3a).

$\begin{matrix}{{U_{1}\left( {i,z} \right)} = {U_{1}^{B} - {\frac{{\Delta\mu}_{Pt}\left( {i,z} \right)}{2F}.}}} & \left( {3a} \right)\end{matrix}$

The term Δμ_(Pt)(i, z) in the above equation (3a) is defined by thefollowing equation (3b).

$\begin{matrix}{{{\Delta\mu}_{Pt}\left( {i,z} \right)} = \frac{\sigma_{Pt}M_{Pt}}{{R\left( {i,z} \right)}_{\rho_{Pt}}}} & \left( {3b} \right)\end{matrix}$

The following equation (4) may be used as a mathematical modelindicative of the rate of platinum oxidation reaction.

$\begin{matrix}{{r_{2}\left( {i,z} \right)} = {k_{2}\begin{bmatrix}{{{\exp \left( {- \frac{{\omega\theta}_{PtO}\left( {i,z} \right)}{RT}} \right)}{\exp \left( {\frac{\alpha_{a,2}n_{2}F}{RT}\left( {E - {U_{2}\left( {i,z} \right)}} \right)} \right)}} -} \\{{\theta_{PtO}\left( {i,z} \right)}\left( \frac{C_{H +}^{2}}{C_{{H +},{ref}}^{2}} \right){\exp \left( {\frac{\alpha_{c,2}n_{2}F}{RT}\left( {E - {U_{2}\left( {i,z} \right)}} \right)} \right)}}\end{bmatrix}}} & (4)\end{matrix}$

The potential U₂(i, z) in the above equation (4) is defined by thefollowing equation (4a).

$\begin{matrix}{{U_{2}\left( {i,z} \right)} = {U_{2}^{\theta} + \frac{{\Delta\mu}_{PtO}\left( {i,z} \right)}{2F} - \frac{{\Delta\mu}_{Pt}\left( {i,z} \right)}{2F}}} & \left( {4a} \right)\end{matrix}$

The term Δμ_(PtO)(i, z) in the above equation (4a) is defined by thefollowing equation (4b).

$\begin{matrix}{{{\Delta\mu}_{PtO}\left( {i,z} \right)} = {{\Delta\mu}_{Pt0}^{0} + \frac{\sigma_{PtO}M_{PtO}}{{R\left( {i,z} \right)}\rho_{PtO}}}} & \left( {4b} \right)\end{matrix}$

The following equation (5) may be used as a mathematical modelindicative of the rate of dissolution of platinum oxide (II).

$\begin{matrix}{{r_{3}\left( {i,z} \right)} = {{k_{3}\left( {{{\theta_{PtO}\left( {i,z} \right)} \cdot \frac{C_{H +}^{2}(z)}{C_{{H +},{ref}}^{2}}} - {\frac{C_{{Pt2} +}(z)}{C_{{{Pt2} +},{ref}}} \cdot \frac{1}{K_{3}\left( {i,z} \right)}}} \right)}.}} & (5)\end{matrix}$

The term K₃(1, z) in the above equation (5) is defined by the followingequation (5a).

$\begin{matrix}{{K_{3}\left( {i,z} \right)} = {\exp \left\lbrack {\frac{F}{RT}\left( {{n_{1}{U_{1}\left( {i,z} \right)}} - {n_{2}{U_{2}\left( {i,z} \right)}}} \right)} \right\rbrack}} & \left( {5a} \right)\end{matrix}$

The following equation (6) is used as a mathematical model of materialbalance when only platinum oxide (II) is taken into consideration.

$\begin{matrix}{\frac{{\theta_{PtO}\left( {i,z} \right)}}{t} = {\left( \frac{{r_{2}\left( {i,z} \right)} - {r_{3}\left( {i,z} \right)}}{\Gamma_{\max}} \right) - {\frac{2{\theta_{PtO}\left( {i,z} \right)}}{R\left( {i,z} \right)} \cdot \frac{{R\left( {i,z} \right)}}{t}}}} & (6)\end{matrix}$

In the above equation (6), Γ_(max) is the maximum surface coverage ofplatinum. The term dR(i, z) in the above equation (6) is defined by thefollowing equation (6a).

$\begin{matrix}{\frac{{R\left( {i,z} \right)}}{t} = {{- \frac{M_{Pt}}{\rho_{Pt}}}\left( {{r_{1}\left( {i,z} \right)} + {r_{2}\left( {i,z} \right)}} \right)}} & \left( {6a} \right)\end{matrix}$

The following equation (7) may be used as a mathematical model ofmaterial balance of platinum ions (II) finally obtained.

$\begin{matrix}{{ɛ\frac{{C_{{Pt2} +}(z)}}{t}} = {{ɛ^{1.5}D_{{Pt2} +}\frac{{C_{{Pt2} +}(z)}}{t}} + {\sum{4\pi \; {R\left( {i,z} \right)}^{2}{N(i)}\left( {{r_{1}\left( {i,z} \right)} + {r_{2}\left( {i,z} \right)}} \right)}}}} & (7)\end{matrix}$

The above-indicated mathematical models may be applied to the fuel cellcatalyst degradation predicting method of the invention, with referenceto, for example, an article of R. M. Darling and J. P. Meyers: J.Electrochem. Soc., vol. 150, pages A1523-A1527, 2003, an article of R.M. Darling and J. P. Meyers: J. Electrochem. Soc., vol. 152; pagesA242-A247, 2005, an article of W. Bi and T. F. Fuller: J. Power Sources,vol. 178, pages 188-196, 2008, etc.

As a first application of the fuel cell catalyst degradation predictingmethod of the invention, this method is used in a control system of afuel cell vehicle. FIG. 3 is a graph schematically showing simulationresults in the first application. In the graph, the horizontal axisindicates the operating time, and the vertical axis indicates theplatinum catalyst surface area. The oval areas in FIG. 3 representapproximate simulation zones in the initial period of the operating timeand the later period of the operating time, respectively. Initially, aparticle size distribution model of a fuel cell catalyst is createdusing the particle size distribution model creating method. Then, usingthe particle size distribution model and the above-indicatedmathematical models, the current surface area of platinum catalystparticles is estimated from physical and chemical data, such as theinitial catalyst particle size, the structure of the catalyst layer, theoperating pattern and operating time up to the present, and so forth. Inthe initial period of the operating time as shown in FIG. 3, the surfacearea of the platinum catalyst particles is sharply reduced. In thisstage, a higher priority is given to prolonging the life of the fuelcell performance, than to improving the fuel efficiency, and control forlowering the upper-limit potential of the fuel cell, or control forshortening the upper-limit potential holding time of the fuel cell areperformed. More specifically, the upper-limit potential of the fuel cellmay be controlled to 1.0V or lower, preferably, 0.9V or lower, morepreferably, 0.85V or lower. In the later period of the operating time asshown in FIG. 3, the surface area of the platinum catalyst particles isreduced gently or at a lower rate in this stage, the upper-limitpotential of the fuel cell is raised in order to improve the fuelefficiency. More specifically, the upper-limit potential of the fuelcell may be controlled to 0.5V or lower, preferably, 0.7V or lower, morepreferably, 0.85V or lower.

As a second application of the fuel cell catalyst degradation predictingmethod of the invention, a fuel cell system may be provided whichinforms the user of the time for maintenance, using information aboutthe estimated condition of catalyst degradation. Initially, a particlesize distribution model of the fuel cell catalyst is created using theparticle size distribution model creating method. Then, using theparticle size distribution model and the above-indicated mathematicalmodels, the current surface area of platinum catalyst particles isestimated from physical and chemical data, such as the initial catalystparticle size, the structure of the catalyst layer, the operatingpattern and operating time up to the present, and so forth. In thesecond application, if the surface area of the platinum catalystparticles is reduced, and reaches a set criterion, the system can informthe user that it is the time for maintenance. Also, in the secondapplication, even if the surface area of the platinum catalyst particlesdoes not reach a set criterion, the system can inform the user how longthe fuel cell can be used before the time for maintenance is reached.

A method of controlling a fuel cell according to the invention ischaracterized by including a step of predicting degradation of the fuelcell catalyst, using the above-described fuel cell catalyst degradationpredicting method, a step of measuring cell voltage of the fuel cell, astep of measuring cell resistance of the fuel cell, a humidity controldetermining step of determining whether humidity control of the fuelcell is to be performed, based on the result of degradation predictionobtained in the fuel cell catalyst deterioration predicting step, avalue of cell voltage obtained in the cell voltage measuring step, and avalue of cell resistance obtained in the cell resistance measuring step,and a step of selecting one of three control modes and performingcontrol of the selected control mode. The three control modes include afirst control mode in which control for reducing the humidity isperformed when it is determined in the humidity control determining stepthat the value of cell resistance obtained in the cell resistancemeasuring step is smaller than a predetermined resistance value, asecond control mode in which control for increasing the humidity of thefuel cell is performed when the value of cell resistance obtained in thecell resistance measuring step is equal to or larger than thepredetermined resistance value, and a third control mode in whichcontrol for increasing the humidity of the fuel cell is performed whenit is predicted from the result of degradation prediction obtained inthe fuel cell catalyst degradation predicting step that the surface areaof the fuel cell catalyst is reduced in a local portion of the catalyst.

Various types of degradation become causes of output reduction of thefuel cell, and an effective control method is different from one type ofdegradation to another. However, the type of degradation cannot bespecified even if the output voltage, output current and resistance aremeasured, as in the related art. In the related art, therefore,adjustment of an output value, or the like, and comparison of the outputwith an output command current value must be continued until the optimumcontrol method is found; therefore, correction of control cannot beeffected with good response.

The causes of reduction of the output performance of the fuel cell areroughly classified into the following three types: (1) reduction of thereaction field due to excessive water caused by reduction of thedraining capability (so-called flooding), (2) reduction of the catalyticactivity due to reduction of the surface area of the platinum catalyst,and (3) increase of resistance to proton transfer due to the presence ofa platinum loss layer. The “platinum loss layer” mentioned in the cause(3) of reduction of the output performance of the fuel cell refers to alayer that is formed in a portion of the catalyst layer close to theelectrolyte membrane as the operating time of the fuel cell increases.The platinum catalyst repeats dissolution and deposition, under theinfluence of potential fluctuations caused by changes in the load of thefuel cell (for example, acceleration and deceleration of a mobile unitwhen the fuel cell is used in the mobile unit). As a result, adegradation phenomenon in the form of a reduction (or loss) of thecatalyst amount in the vicinity of the electrolyte membrane occurs. Theplatinum loss layer is framed as a result of the degradation phenomenon.Since the platinum loss layer is formed in a portion of the catalystlayer close to the electrolyte membrane, the proton transfer distance inthe catalyst layer is extended or increased, resulting in an increase ofresistance to proton transfer, and reduction of the performance of thefuel cell.

In the case where the reduction of the output performance of the fuelcell is due to the cause (1) as described above (namely, in the case offlooding), for example, when the water-repellent property of carbon thatsupports platinum deteriorates, and water formed in the cell is lesslikely to be discharged from the membrane electrode assembly, the watermay reduce the reaction field in the cathode, thus causing a reductionof the cell voltage. In this case, control for reducing the humidity(e.g., control for increasing the amount of reaction gas supplied) iseffectively performed. In the case where the reduction of the outputperformance of the fuel cell is due to the cause (2) as described above,there is no need to perform control concerning the humidity. In the casewhere the reduction of the output performance of the fuel cell is due tothe cause (3) as described above, control for increasing the humidity ofthe fuel cell (e.g., control for humidifying the fuel cell with ahumidifying module, control for lowering the gas inflow pressure,control for reducing the amount of reaction gas supplied) is effectivelyperformed so as to prevent reduction of the cell voltage. Thus, it isnecessary to determine the necessity to control the humidity and controlthe humidity if necessary, with respect to each of the causes (1), (2),(3) of the reduction of the fuel cell output performance.

Although the cause (1) of the reduction of the fuel cell outputperformance is detected only through measurements of the output voltageand output current, it is extremely difficult to detect the causes (2)and (3) of the reduction of the fuel cell output performance whiledistinguishing them from each other, since the cell resistance does notchange in either of the cases where the causes (2) and (3) occur. Thecell resistance does not change for the following two reasons: (1) ifeven a single particle of the platinum catalyst particles remains in theplatinum loss layer, an electron conduction path is formed, and the cellresistance does not change (this situation does not appear in DCresistance), and (2) if there is a humidity distribution in the cellplane, the resistance of a portion having the lowest resistance in theplane is reflected by the cell resistance.

FIG. 4 is a graph showing examples of calculation results indicatingprediction results obtained in the fuel cell catalyst degradationpredicting step of the invention. In the graph of FIG. 4, the verticalaxis indicates the platinum catalyst surface area per unit thickness ofthe catalyst layer, and the horizontal axis indicates or specifies theposition taken in the thickness direction of the catalyst layer wherethe position of the interface with the electrolyte membrane is denotedas 0, and the position of the interface with the gas diffusion layer isdenoted as 1. In FIG. 4, black diamond plots indicate surface areas ofthe platinum catalyst at respective positions in the initial period ofthe operating time, and white square plots, black triangle plots, whitecircle plots, and black square plots indicate surface areas of theplatinum catalyst after a lapse of a certain amount of time the fuelcell is used, in increasing order of the operating time of the fuelcell. Also, in the examples of the calculation results, a thresholdvalue of platinum catalyst surface area is determined in advance, and aportion of the catalyst layer below the threshold value is regarded as aplatinum loss layer, as shown in FIG. 4. As is understood from FIG. 4,the platinum catalyst surface area that is kept at an equally high valuein the catalyst layer during the initial period of the operating time(black diamond plots) starts rapidly decreasing, particularly at aroundthe interface with the electrolyte membrane, as the operating timeincreases (white square plots→black triangle plots→white circleplots→black square plots). As the platinum catalyst surface areadecreases, the thickness of the platinum loss layer rapidly increases ataround the interface with the electrolyte membrane. Calculation forestimating the platinum catalyst surface area distribution in thethickness direction of the catalyst layer may be conducted each time agiven operating time elapses (for, example, once every 100 hours).Namely, the calculation is not necessarily required for each operation.A resistance overvoltage is developed depending on the thickness of theplatinum loss layer. The resistance overvoltage may be calculated inadvance, using a formula like the following equation (8), for example,and may be used for feedback control.

ΔW=I ² ×R _(Pt-loss) ×L  (8)

(where, ΔW: output reduction (W) caused by the platinum loss layer, I:current (A), R_(Pt-loss): resistance (Ω/cm²) per unit area of theplatinum loss layer, and L: thickness (cm) of the platinum loss layer.)

The method of controlling the fuel cell according to the invention hasthe fuel cell catalyst deterioration predicting step, cell voltagemeasuring step, cell resistance measuring step, step of determiningwhether humidity control is to be performed (which will be called“humidity control execution determining step”), and a step (which willbe called “humidity control executing step”) of effecting a selected oneof different control modes, based on the result of determinationobtained in the humidity control execution determining step. Of thesesteps, the fuel cell catalyst degradation predicting step has alreadybeen described above in the explanation of the fuel cell catalystdegradation predicting method of the invention. Also, known methods ofthe related art may be used as a method of measuring the cell voltage ofthe fuel cell in the cell voltage measuring step and a method ofmeasuring the cell resistance of the fuel cell in the cell resistancemeasuring step. In the following, the humidity control executiondetermining step and the humidity control executing step will bedescribed in detail.

In the humidity control execution determining step of the invention, itis determined whether humidity control is to be performed, based on theresult of degradation prediction obtained in the fuel cell catalystdegradation predicting step, the value of cell voltage obtained in thecell voltage measuring step, and the value of cell resistance obtainedin the cell resistance measuring step. For example, the measured cellvoltage V is compared with a predetermined threshold value V₁. If V<V₁,for example, it is found that some abnormality occurs in the fuel cell,and the performance of the fuel cell is reduced due to the abnormality.Also, the measured cell resistance R is compared with a predetermined R₁(the minimum value in a permissible range of cell resistance). If R<R₁,it may be determined that flooding occurs (which is the cause (I) ofreduction of the output performance of the fuel cell), and that controlfor reducing the humidity needs to be performed. Furthermore, themeasured cell resistance R is compared with a predetermined R₂ (themaximum value in the permissible range of cell resistance). If R>R₂, itmay be determined that the increase of the resistance value occurs dueto drying or low humidity, and that control for increasing the humidityneeds to be performed. Also, if a reduction of the surface area in alocal portion of the fuel cell catalyst can be predicted from the resultof prediction of degradation obtained in the fuel cell catalystdegradation predicting step, it may be determined that voltage reductionoccurs due to the presence of the platinum loss layer (the cause (3) ofreduction of the output performance of the fuel cell), and it may bedetermined that control for increasing the humidity needs to beperformed. In contrast to this case, if it can be predicted that thesurface area of the fuel cell catalyst is not locally reduced, it may bedetermined that voltage reduction occurs due to reduction of the surfacearea of the platinum catalyst (the cause (2) of reduction of the outputperformance of the fuel cell), and it may be determined that humiditycontrol is not particularly needed. While it may be determined whetherhumidity control needs to be performed, by individually evaluating thedegradation prediction result obtained in the fuel cell catalystdegradation predicting step, the value of cell voltage obtained in thecell voltage measuring step, and the value of cell resistance obtainedin the cell resistance measuring step, these prediction result andmeasurement values may be comprehensively evaluated or assessed so as todetermine whether the humidity control needs to be performed.

In the humidity control executing step of the invention, a control modemay be selected from at least the first control mode in which controlfor reducing the humidity is performed, and the second and third controlmodes in which control for increasing the humidity is performed,according to the circumstances, and the selected control mode may becarried out. A specific method of controlling the humidity is notparticularly limited. Rather, different control methods may be useddepending on the control mode selected, or, even in the same controlmode, the humidity may be controlled by different control methodsaccording to the circumstances. Specific examples of the method ofcontrolling the humidity include, for example, control using ahumidifying module, control of the gas inflow pressure, control of theamount of reaction gas supplied, and control of the flow rate of thecoolant. In particular, the coolant flow rate control is performed so asto control the cell temperature, thereby to indirectly control thehumidity. In the first control mode, at least one control selected fromcontrol for increasing the gas inflow pressure, control for increasingthe amount of reaction gas supplied, and control for reducing the flowrate of the coolant (namely, control for raising the cell temperature)is performed so as to reduce the humidity. In the second and thirdcontrol modes, at least one control selected from control for reducingthe gas inflow pressure, control for reducing the amount of reaction gassupplied, and control for increasing the flow rate of the coolant(namely, control for lowering the cell temperature) is performed so asto increase the humidity.

FIG. 5 is a flowchart illustrating a typical example of the method ofcontrolling the fuel cell according to the invention. Initially, thecell voltage V is measured in step S11 as shown in FIG. 5, and themeasured cell voltage V is compared with a predetermined threshold valueV₁ in step S12. The control ends if V is equal to or larger than V_(I),and the control proceeds to step S13 if V is smaller than V₁. The cellresistance is measured in step S13, and the measured cell resistance Ris compared with a predetermined threshold value R₁ (the maximum valuein a permissible range of the cell resistance) in step S14. If R issmaller than R₁, it is determined that flooding occurs (which is thecause (1) of reduction of the output performance of the fuel cell), andthe first control mode is selected in which control for reducing thehumidity is performed (step S15). After the control for reducing thehumidity is performed, the control returns to step S13, and the firstcontrol mode is selected to keep the humidity reduction controlperformed until the resistance value R becomes equal to or larger thanR₁. If it is determined in step S14 that R is equal to or larger thanR₁, the control proceeds to step S16. In step S16, the measured cellresistance R is compared with a predetermined threshold value R₂ (themaximum value in the permissible range of cell resistance). If R islarger than R₂, the second control mode is selected in which control forincreasing the humidity is performed (step S17). After the control forincreasing the humidity is performed, the control returns to step S13,and the second control mode is selected to keep the control forincreasing the humidity performed until the resistance value R becomesequal to or larger than R₁ and equal to or smaller than R₂ (R₁≦R≦₂). Ifit is determined in step S16 that R is equal to or smaller than R₂, thecontrol proceeds to step S18. In step S18, it is predicted whether thesurface area of a portion of the platinum catalyst close to theelectrolyte membrane is reduced, using the degradation prediction resultobtained in the fuel cell catalyst degradation predicting step. For theprediction, the graph of examples of calculation results as shown inFIG. 4 may be used. If it is predicted from the graph that the surfacearea of the fuel cell catalyst is locally reduced, more specifically,the surface area of a portion of the platinum catalyst close to theelectrolyte membrane is reduced, it is determined that the voltagereduction is due to the presence of the platinum loss layer (which isthe cause (3) of reduction of the output performance of the fuel cell),and the third control mode in which control for increasing the humidityis performed is selected so as to increase the humidity (step 521). Ifit is predicted that the surface area of the fuel cell is not locallyreduced, more specifically, the surface area of a portion of theplatinum catalyst close to the electrolyte membrane is not reduced, itis determined that the voltage reduction is due to reduction of thesurface area of the platinum catalyst (which is the cause (2) ofreduction of the output performance of the fuel cell), and no particularcontrol concerning the humidity is performed (step S20). After executionof step S20 or step S21, the system control ends.

While specific embodiments of the invention will be described in furtherdetail, it is to be understood that the invention is not limited tothese embodiments, but may be otherwise embodied without departing fromthe principle of the invention.

A particle size distribution model was created, using the particle sizedistribution model creating method of the invention. To create themodel, Microsoft Office Excel (trade name, manufactured by Microsoft)was used as spreadsheet software, and the NORMINV function that returnsvalues of the inverse function of the cumulative distribution functionof a normal distribution with respect to the designated average andstandard deviation was used as the function. The NORMINV function hasthree arguments of NORMINV (probability x, average μ, standard deviationσ).

To create a particle size distribution model 1, a particle size rangewas determined in which the minimum particle size is 1.9 (in nm, thesame unit will be used for other particle sizes), and the maximumparticle size is 3.1. The average particle size was 2.5. Then, thedetermined particle range was divided into 10 regions. Morespecifically, x=0.1, 0.2, 0.3, . . . 0.9, μ=2.5, and standard deviationσ=0.2 were respectively substituted into the NORMINV function, and 9division points were determined by calculating a value of F(X)⁻¹) foreach probability, namely, values of the inverse function of thecumulative distribution function. The results of calculation areindicated in TABLE 2 below.

TABLE 2 x F(x)⁻¹ 0.1 2.2437 0.2 2.3317 0.3 2.3951 0.4 2.4493 0.5 2.50000.6 2.5507 0.7 2.6049 0.8 2.6683 0.9 2.7563

The values of 1.9 (minimum particle size) and 3.1 (maximum particlesize) were added to the values of these division points, so that a totalof 11 typical points were determined. With the thickness in the actualcatalyst layer further taken into consideration, the particle sizedistribution model 1 was created. In this example, 10 layers of theparticle size distribution models each having particles sizes of theabove-indicated 11 typical points were superimposed in the direction ofthickness of the catalyst layer, to create the model (TABLE 3). Namely,this model has a uniform particle size distribution as viewed in thedirection of thickness of the catalyst layer.

TABLE 3 z (Mesh in Thickness Direction of Catalyst Layer 1 2 3 4 5 6 7 89 10 1 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 1.9 2 2.2437 2.2437 2.24372.2437 2.2437 2.2437 2.2437 2.2437 2.2437 2.2437 3 2.3317 2.3317 2.33172.3317 2.3317 2.3317 2.3317 2.3317 2.3317 2.3317 4 2.3951 2.3951 2.39512.3951 2.3951 2.3951 2.3951 2.3951 2.3951 2.3951 5 2.4493 2.4493 2.44932.4493 2.4493 2.4493 2.4493 2.4493 2.4493 2.4493 6 2.5000 2.5000 2.50002.5000 2.5000 2.5000 2.5000 2.5000 2.5000 2.5000 7 2.5507 2.5507 2.55072.5507 2.5507 2.5507 2.5507 2.5507 2.5507 2.5507 8 2.6049 2.6049 2.60492.6049 2.6049 2.6049 2.6049 2.6049 2.6049 2.6049 9 2.6683 2.6683 2.66832.6683 2.6683 2.6683 2.6683 2.6683 2.6683 2.6683 10 2.7563 2.7563 2.75632.7563 2.7563 2.7563 2.7563 2.7563 2.7563 2.7563 11 3.1 3.1 3.1 3.1 3.13.1 3.1 3.1 3.1 3.1

To create a particle size distribution model 2, a particle size rangewas determined in which the minimum particle size is 1.4, and themaximum particle size is 2.6. The average particle size was 2. Then, thedetermined particle range was divided into 10 regions. Morespecifically, x=0.1, 0.2, 0.3, . . . 0.9, μ=2 (in nm, a preferable valueas the catalyst particle size), and standard deviation σ=0.2 wererespectively substituted into the NORMINV function, and 9 divisionpoints were determined by calculating a value of F(X)⁻¹ for eachprobability, namely, values of the inverse function of the cumulativedistribution function. The results of calculation are indicated in TABLE4 below.

TABLE 4 x F(X)⁻¹ 0.1 1.7436 0.2 1.8316 0.3 1.8952 0.4 1.9494 0.5 2 0.62.0506 0.7 2.1048 0.8 2.1684 0.9 2.2564

Also, the determined particle size range was divided into 80 regions.More specifically, x=0.0125, 0.0250, 0.0375, . . . 0.9875, μ=2 (in nm, apreferable value as the catalyst particle size), and standard deviationσ=0.2 were respectively substituted into the NORMINV function, and 79division points were determined by calculating a value of F(X)⁻¹ foreach probability, namely, values of the inverse function of thecumulative distribution function. The results of calculation areindicated in TABLE 5 below.

TABLE 5 x F(X)⁻¹ 0.0125 1.5517 0.0250 1.6080 0.0375 1.6439 0.0500 1.67100.0625 1.6932 0.0750 1.7121 0.0875 1.7287 0.1000 1.7437 0.1125 1.75730.1250 1.7699 0.1375 1.7817 0.1500 1.7927 0.1625 1.8032 0.1750 1.81310.1875 1.8226 0.2000 1.8317 0.2125 1.8404 0.2250 1.8489 0.2375 1.85710.2500 1.8651 0.2625 1.8729 0.2750 1.8804 0.2875 1.8879 0.3000 1.89510.3125 1.9022 0.3250 1.9092 0.3375 1.9161 0.3500 1.9229 0.3625 1.92960.3750 1.9363 0.3875 1.9428 0.4000 1.9493 0.4125 1.9558 0.4250 1.96220.4375 1.9685 0.4500 1.9749 0.4625 1.9812 0.4750 1.9875 0.4875 1.99370.5000 2.0000 0.5125 2.0063 0.5250 2.0125 0.5375 2.0188 0.5500 2.02510.5625 2.0315 0.5750 2.0378 0.5875 2.0442 0.6000 2.0507 0.6125 2.05720.6250 2.0637 0.6375 2.0704 0.6500 2.0771 0.6625 2.0839 0.6750 2.09080.6875 2.0978 0.7000 2.1049 0.7125 2.1121 0.7250 2.1196 0.7375 2.12710.7500 2.1349 0.7625 2.1429 0.7750 2.1511 0.7875 2.1596 0.8000 2.16830.8125 2.1774 0.8250 2.1869 0.8375 2.1968 0.8500 2.2073 0.8625 2.21830.8750 2.2301 0.8875 2.2427 0.9000 2.2563 0.9125 2.2713 0.9250 2.28790.9375 2.3068 0.9500 2.3290 0.9625 2.3561 0.9750 2.3920 0.9875 2.4483

Finally, with regard to the calculation results of TABLE 4 in which thedetermined particle size range was divided into 10 regions, the regionincluding the maximum particle size, namely, the section where the valueof the particle size is between 2.2564 and 2.6, was replaced by a part(data in the form of boldface and underlined numbers in TABLE 5) of thecalculation results of TABLE 5 in which the determined particle sizerange was divided into 80 regions. Namely, through this operation, theparticle size range from 1.4 as the minimum particle size to 2.6 as themaximum particle size was divided into 10 regions, and the region orrange including the maximum particle size was further divided into 8regions. Accordingly, a total of 16 division points were determined asshown in TABLE 6 below.

TABLE 6 x F(X)⁻¹ 0.1 1.7436 0.2 1.8316 0.3 1.8952 0.4 1.9494 0.5 2 0.62.0506 0.7 2.1048 0.8 2.1684 0.9 2.2564 0.9125 2.2713 0.925 2.28790.9375 2.3068 0.95 2.329 0.9625 2.3561 0.975 2.392 0.9875 2.4483

The values of 1.4 (minimum particle size) and 2.6 (maximum particlesize) were added to the values of these division points, so that a totalof 18 typical points were determined. Thus, the particle distributionmodel 2 containing particles having the particle sizes of the typicalpoints was created.

Using the method of predicting degradation of the fuel cell catalystaccording to the invention, the platinum surface area maintenance factorwith respect to the operating time (the number of power generationcycles) was calculated. In this embodiment, the particle sizedistribution model as shown in TABLE 3 was used as a particle sizedistribution model of the platinum catalyst in the initial state. FIG.6A is a schematic view of the distribution mode. For calculation of theplatinum surface area maintenance factor, mathematical software (gPROMS)was used. By entering various parameters including equations, boundaryconditions, the initial catalyst particle size, the structure of thecatalyst-layer, such as the thickness of the catalyst layer, and theoperating pattern, numerical solutions that vary with time wereautomatically obtained. In this manner, a graph of the platinum surfacearea maintenance factor with respect to the operating time (the numberof power generation cycles) as shown in FIG. 6B was obtained. Thecalculation time was about 20 minutes.

As a comparative example, a particle size distribution model in whichthe same difference in the particle size is taken as equal intervals andthe number of particles is a variable was created, separately from theparticle size distribution models of the above embodiments of theinvention. TABLE 7 below indicates the particle sizes (nm) of theparticle size distribution model of the comparative example, and theratio of the number of particles having one of these particle sizes to atotal number of the particles. FIG. 8A is a schematic view of thedistribution model. Using the particle size distribution model as shownin FIG. 8A, degradation of the fuel cell catalyst was predicted, and theplatinum surface area maintenance factor with respect to the operatingtime (the number of power generation cycles) was calculated.

TABLE 7 Particle Size (nm) Ratio 2.0 0.0026613 2.1 0.0134476 2.20.0474085 2.3 0.1166061 2.4 0.2000968 2.5 0.2395594 2.6 0.2000968 2.70.1155061 2.8 0.0474085 2.9 0.0134476 3.0 0.0026613

For calculation of the platinum surface area maintenance factor,

mathematical software (gPROMS) was used. By entering various parametersincluding equations, boundary conditions, the initial catalyst particlesize, the structure of the catalyst layer, such as the thickness of thecatalyst layer, and the operating pattern, numerical solutions that varywith time were automatically obtained. In this manner, a graph of theplatinum surface area maintenance factor with respect to the operatingtime (the number of power generation cycles) as shown in FIG. 8B wasobtained. The calculation time was about 20 minutes.

As is understood from FIG. 6B, the calculation results (solid line inFIG. 6B) of the embodiment of the invention almost precisely simulatethe experimental results (plots on the graph); it is thus found that themethod of predicting degradation of the fuel cell catalyst according tothe invention makes it possible to precisely simulate degradation of thefuel cell catalyst which would occur in reality. In contrast to this, asis understood from FIG. 8B, the calculation results (solid line in FIG.8B) of the comparative example exhibit poor preciseness in thesimulation of the experimental results (plots on the graph), as comparedwith the calculation results of the embodiment of the invention.

1. A particle size model creating method of creating a particle sizedistribution model that simulates a particle size distribution of acluster of particles of a catalyst metal of a fuel cell, said cluster ofparticles comprising a plurality of particles of the catalyst metal,comprising: determining a particle size range by determining a minimumparticle size and a maximum particle size of the cluster of particles ofthe catalyst metal to be simulated; integrating the frequency ofappearance of the particles in the determined particle size range, overan integration region defined by the minimum particle size as a startingpoint and the maximum particle size as an endpoint; dividing theintegration region into a given number of regions through a firstdividing operation, using the integral of the frequency of appearance,so that integrals obtained for the individual regions into which theintegration region is divided are substantially equal, and determiningparticles sizes, of division points at which the integration region isdivided; determining the minimum particle size, the maximum particlesize and the particle sizes of the respective division points, astypical points; and obtaining a particle size distribution model byassuming a particle size distribution containing particles having theparticle sizes of the respective typical points, said particle sizedistribution being plotted such that the frequency of appearance of theparticles having the particle size of each of the typical points isequal to the integral obtained for each of the regions into which theintegration region is divided at the typical points.
 2. The particlesize distribution model creating method according to claim 1, furthercomprising dividing a part of or all of the given number of regionsobtained through the first dividing operation, further into a givennumber of regions, through a second dividing operation, so thatintegrals obtained for the individual regions resulting from the seconddividing operation are substantially equal, and determining particlesizes that provide division points in the first dividing operation andthe second dividing operation, as the typical points.
 3. The particlesize distribution model creating method according to claim 2, whereinthe second dividing operation is performed on a region including themaximum particle size, out of the given number of regions resulting fromthe first dividing operation.
 4. The particle size distribution modelcreating method according to claim 1, wherein the frequency ofappearance is one of the number of particles, mass and volume of thecatalyst metal.
 5. The particle size distribution model creating methodaccording to claim 1, wherein the particle distribution is one of anormal distribution, an exponential distribution, a t-distribution, achi-square distribution, a gamma distribution, a beta distribution, anF-distribution, a Cauchy distribution, an Erlang distribution, atriangular distribution, a Laplace distribution, a Rayleighdistribution, a logistic distribution, a Pareto distribution, a Weibulldistribution, and functions referring to an actually measured platinumparticle size distribution.
 6. A degradation predicting method ofpredicting degradation of a catalyst metal of a fuel cell, comprising:determining a particle size range by determining a minimum particle sizeand a maximum particle size of a cluster of particles of the catalystmetal of which a particle size distribution is to be simulated, saidcluster of particles comprising a plurality of particles of the catalystmetal; integrating the frequency of appearance of the particles in thedetermined particle size range, over an integration region defined bythe minimum particle size as a starting point and the maximum particlesize at an endpoint; dividing the integration region into a given numberof regions through a first dividing operation, using the integral of thefrequency of appearance, so that integrals obtained for the individualregions into which the integration region is divided are substantiallyequal, and determining particles sizes of division points at which theintegration region is divided; determining the minimum particle size,the maximum particle size and the particle sizes of the respectivedivision points, as typical points; obtaining a particle sizedistribution model by assuming a particle size distribution containingparticles having the particle sizes of the respective typical points,said particle size distribution being plotted such that the frequency ofappearance of the particles having the particle size of each of thetypical points is equal to the integral obtained for each of the regionsinto which the integration region is divided at the typical points; andpredicting degradation of the catalyst metal of the fuel cell, using theparticle size distribution model.
 7. The degradation predicting methodaccording to claim 6, wherein degradation of the catalyst metal of thefuel cell is predicted, using at least one mathematical model selectedfrom a mathematical model indicative of the rate of dissolution reactionof the catalyst metal, a mathematical model indicative of the rate ofoxidation reaction of the catalyst metal, and a mathematical modelindicative of material balance of the catalyst metal.
 8. The degradationpredicting method according to claim 6, wherein: the catalyst metal ofthe fuel cell is platinum; and degradation of platinum of the fuel cellis predicted, using all of a mathematical model indicative of the rateof dissolution reaction of platinum, a mathematical model indicative ofthe rate of oxidation reaction of platinum, a mathematical modelindicative of the rate of dissolution reaction of platinum oxide (II),and a mathematical model indicative of material balance.
 9. A method ofcontrolling a fuel cell, comprising: determining a particle size rangeby determining a minimum particle size and a maximum particle size of acluster of particles of the catalyst metal of which a particle sizedistribution is to be simulated, said cluster of particles comprising aplurality of particles of the catalyst metal; integrating the frequencyof appearance of the particles in the determined particle size range,over an integration region defined by the minimum particle size as astarting point and the maximum particle size as an endpoint; dividingthe integration region into a given number of regions through a firstdividing operation, using the integral of the frequency of appearance,so that integrals obtained for the individual regions into which theintegration region is divided are substantially equal, and determiningparticles sizes of division points at which the integration region isdivided; determining the minimum particle size, the maximum particlesize and the particle sizes of the respective division points, astypical points; obtaining a particle size distribution model by assuminga particle size distribution containing particles having the particlesizes of the respective typical points, said particle size distributionbeing plotted such that the frequency of appearance of the particleshaving the particle size of each of the typical points is equal to theintegral obtained for each of the regions into which the integrationregion is divided at the typical points; predicting degradation of thecatalyst metal of the fuel cell, using the particle size distributionmodel; measuring a cell voltage of the fuel cell; measuring a cellresistance of the fuel cell; determining whether humidity control of thefuel cell is to be performed, based on a degree of the predicteddegradation of the catalyst metal of the fuel cell, the measured cellvoltage, and the measured cell resistance; and executing a first controlmode in which control for reducing the humidity of the fuel cell isperformed when the measured cell resistance is smaller than apredetermined resistance value, a second control mode in which controlfor increasing the humidity of the fuel cell is performed when themeasured cell resistance is equal to or larger than the predeterminedresistance value, and a third control mode in which control forincreasing the humidity of the fuel cell is performed when it ispredicted, from a result of degradation prediction of the catalyst metalof the fuel cell, that a surface area of a local portion of the fuelcell catalyst is reduced.
 10. The method of controlling a fuel cell,according to claim 9, wherein: at least one of control for increasing apressure under which at least one of fuel gas and oxidizing gas issupplied to the fuel cell, control for increasing the amount of at leastone of the fuel gas and oxidizing gas supplied to the fuel cell, andcontrol for reducing the flow rate of a coolant of the fuel cell isperformed in the first control mode; at least one of control forreducing a pressure under which at least one of the fuel gas andoxidizing gas is supplied to the fuel cell, control for reducing theamount of at least one of the fuel gas and oxidizing gas supplied to thefuel cell, and control for increasing the flow rate of the coolant ofthe fuel cell is performed in one of the second control mode and thethird control mode.